Representations of the Hyperalgebra of an Algebraic Group

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.Introduction. We study some aspects of the question of the rationality of finite-dimensional representations of the hyperalgebra of an affine algebraic group scheme G. This is a question of Verma's*, according to [1].We will look at non-rational representations of the hyperalgebras of the additive and multiplicative groups. On the other hand, we show that for simply connected, connected semisimple groups at least the simple representations of the hyperalgebra are rational (see Section 5 for the precise results). Let G be a reduced affine algebraic group scheme over the prime field Fp of characteristic p, represented by the Hopf algebra A. Let M be the augmentation ideal of A, and suppose that G is connected in the sense that nM(P n)A= (0), where MPpn) =f{aPna E M}. The family {A/M(P )A}n>0 of finite dimensional Hopf algebras, together with the quotient morphisms A/M(Pn)A-)A/M(P )A for e < n, is a projective family. Definition. A is the projective limithhm Mp -A/M(P )A, n in the category of Hopf algebras (h lim is used to indicate the category in which the projective limit lies). Let G be the group scheme represented by A. G plays a role in the representation theory of the hyperalgebra of G. The dual family {(A/M(Pn)A)*}n>0 of Hopf algebras is an inductive family on a directed set.

Simply Connected Groups, the Hyperalgebra, and Verma's Conjecture

“…The proof of the second fact completes an aspect of the work begun in the paper [1] on the hyperalgebra. In characteristic zero, a group is called simply connected if it cannot be covered by another group by a map with finite, non-trivial kernel.…”

“…In order to get a complete picture of the representation theory of the group in terms of that of the hyperalgebra, one questions whether every finite-dimensional representation of the hyperalgebra comes from a representation of the group (see [1] for instance).…”

Rational Cohomology and Supports for Linear Algebraic Groups

“…The following theorem of J. Sullivan in [56] reveals the close connection of the representation theory of the family {G (r) , r > 0} of algebras with the rational representations of G. Recall that if A is a k-algebra and M is an A-module, then M is said to be locally finite if each finite dimensional subspace of M is contained in some finite dimensional A-submodule of M . Theorem 2.16.…”

Relations Between the Cohomology of an Algebraic Group and Its Infinitesimal Subgroups

“…Let G= lim G be the inductive limit of the groups G' in the category of affine k-groups (see [3] and [4].) Let A be the coordinate ring of G. There is a canonical injective mapping A _+A corresponding to the epimorphism G-G induced by is the C-linear map such that (Ad( g)(l ))(a ?1 ) = 2 l(a(2))g(a(j))g(S(a(3))), for a E A and 1 E L.…”